7.3 Strategies, Semi-martingales and Stochastic Integration 1177.3 Strategies, Semi-martingales and Stochastic Integration
Integration
The simplest strategy an agent can follow, is to buy at a deterministic time
T 1 ∈Rand sell at a later timeT 2 ≥T 1 ,T 2 ∈R. This situation was already
encountered in Chap. 2 and further developed in Chaps. 5 and 6. So let us
discuss this elementary strategy of buying and selling. To make decisions
possible the random timesT 1 andT 2 can only depend on past information and
therefore need to bestopping times. Since we can give “limits” to our broker,
the decision to buy/sell at timeT 1 can depend on information available at time
T 1. Therefore the number of assets we buy at timeT 1 should be measurable
with respect toFT 1. By acting in such a way the agent holdsf:Ω→Rdassets
from timeT 1 to timeT 2 ,wherefis anFT 1 -measurableRd-valued function.
This action results in a gain (or loss) equal to (f, ST 2 −ST 1 ).
Definition 7.3.1.A predictable processH:R+×Ω→RdwithH 0 =0is
said to be
(i) a simple strategy if there are stopping times 0 ≤T 0 ≤T 1 ···≤Tn<∞,as
well as random variablesf 0 ,···,fn− 1 ,whereeachfkisFTk-measurable,
such thatH=
∑n− 1
k=0fk^1 ]]Tk,Tk+1]],
(ii) a bounded simple strategy if in addition,f 0 ,···,fn− 1 are inL∞,
(iii)of bounded support if there is a real numbert∈R+such thatH=H (^1) [[ 0,t]].
IfHis a simple strategy then the ultimate gain equals
(H·S)∞:=
n∑− 1k=0(
fk,STk+1−STk)
(7.4)
and at timetthe portfolio has a gain equal to
(H·S)t:=n∑− 1k=0(
fk,STk+1∧t−STk∧t)
. (7.5)
The processH·Sis called the stochastic integral ofHwith respect toS.It
has to be seen as a process. The ultimate gain is described by the random
variable (H·S)∞= limt→∞(H·S)t(where the limit trivially exists). Another
notation is
H·S=∫
HudSu (7.6)Summing up, the definition of a stochastic integral forsimpleintegrands
goes exactly along the lines of the setting of finite discrete time as encountered
in Chaps. 2, 5 and 6. There is no limiting procedure involved so far: the
integrals (7.4), (7.5) and (7.6) reduce to finite sums.
The crucial step now consists in extending this notion from simple inte-
grands to more general ones by an appropriate limiting procedure. This is